Properties

Label 2-210-105.2-c1-0-10
Degree $2$
Conductor $210$
Sign $0.663 - 0.748i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)2-s + (1.53 − 0.796i)3-s + (−0.866 + 0.499i)4-s + (0.136 + 2.23i)5-s + (1.16 + 1.27i)6-s + (2.35 − 1.20i)7-s + (−0.707 − 0.707i)8-s + (1.73 − 2.45i)9-s + (−2.12 + 0.709i)10-s + (−3.01 + 1.73i)11-s + (−0.933 + 1.45i)12-s + (−0.354 + 0.354i)13-s + (1.77 + 1.96i)14-s + (1.98 + 3.32i)15-s + (0.500 − 0.866i)16-s + (−3.12 − 0.838i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.887 − 0.459i)3-s + (−0.433 + 0.249i)4-s + (0.0611 + 0.998i)5-s + (0.476 + 0.522i)6-s + (0.889 − 0.456i)7-s + (−0.249 − 0.249i)8-s + (0.577 − 0.816i)9-s + (−0.670 + 0.224i)10-s + (−0.908 + 0.524i)11-s + (−0.269 + 0.421i)12-s + (−0.0982 + 0.0982i)13-s + (0.474 + 0.523i)14-s + (0.513 + 0.858i)15-s + (0.125 − 0.216i)16-s + (−0.758 − 0.203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.663 - 0.748i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.663 - 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54151 + 0.693109i\)
\(L(\frac12)\) \(\approx\) \(1.54151 + 0.693109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (-1.53 + 0.796i)T \)
5 \( 1 + (-0.136 - 2.23i)T \)
7 \( 1 + (-2.35 + 1.20i)T \)
good11 \( 1 + (3.01 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.354 - 0.354i)T - 13iT^{2} \)
17 \( 1 + (3.12 + 0.838i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.42 - 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.64 + 1.51i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 1.97T + 29T^{2} \)
31 \( 1 + (4.44 + 7.69i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.09 - 2.16i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.96iT - 41T^{2} \)
43 \( 1 + (6.03 - 6.03i)T - 43iT^{2} \)
47 \( 1 + (-0.617 - 2.30i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.55 + 5.79i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.18 + 3.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.79 + 6.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.64 - 13.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.96iT - 71T^{2} \)
73 \( 1 + (-3.68 - 0.988i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.48 - 4.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.4 - 10.4i)T + 83iT^{2} \)
89 \( 1 + (3.59 - 6.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.6 - 11.6i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90886497201501633669934206088, −11.55604942612676066464689374806, −10.48196539442343166513024861087, −9.387007772884094658943338945446, −8.163240609602049448356189725122, −7.40482898601925816084509820082, −6.75272730054283949462377574083, −5.13947286879923261317874217134, −3.72574197089088489131344083334, −2.26920789853026370567478421808, 1.81155972671823520741813135596, 3.25077983589614827591850155066, 4.76712902690899456221339040858, 5.31566005833493545106469421807, 7.58348952998528700916300350937, 8.712195007701393487005134843329, 9.017774191910735502887887431986, 10.37068533036564522394327047898, 11.18924073881005750335150595305, 12.33498802565273358709503975550

Graph of the $Z$-function along the critical line